Optimal. Leaf size=40 \[ \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^2}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A] time = 0.09, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2094, 208} \[ \frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^2}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2094
Rubi steps
\begin {align*} \int \frac {x \left (2 e-2 f x^3\right )}{e^2+4 e f x^3-4 d f x^4+4 f^2 x^6} \, dx &=-\left (\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{e^2-16 d e^2 f x^2} \, dx,x,\frac {x^2}{-2 e-4 f x^3}\right )\right )\\ &=\frac {\tanh ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^2}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 86, normalized size = 2.15 \[ -\frac {\text {RootSum}\left [4 \text {$\#$1}^6 f^2-4 \text {$\#$1}^4 d f+4 \text {$\#$1}^3 e f+e^2\& ,\frac {\text {$\#$1}^3 f \log (x-\text {$\#$1})-e \log (x-\text {$\#$1})}{6 \text {$\#$1}^4 f-4 \text {$\#$1}^2 d+3 \text {$\#$1} e}\& \right ]}{2 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 155, normalized size = 3.88 \[ \left [\frac {\sqrt {d f} \log \left (\frac {4 \, f^{2} x^{6} + 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2} + 4 \, {\left (2 \, f x^{5} + e x^{2}\right )} \sqrt {d f}}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {-d f} \arctan \left (\frac {\sqrt {-d f} x}{d}\right ) - \sqrt {-d f} \arctan \left (\frac {{\left (2 \, f x^{4} - 2 \, d x^{2} + e x\right )} \sqrt {-d f}}{d e}\right )}{2 \, d f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {2 \, {\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 74, normalized size = 1.85 \[ -\frac {\left (\RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right )^{4} f -\RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right ) e \right ) \ln \left (-\RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right )+x \right )}{2 f \left (6 f \RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right )^{5}-4 d \RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right )^{3}+3 e \RootOf \left (4 f^{2} \textit {\_Z}^{6}-4 d f \,\textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{3}+e^{2}\right )^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, \int \frac {{\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.38, size = 67, normalized size = 1.68 \[ -\frac {\mathrm {atanh}\left (\frac {27\,e^2\,\sqrt {f}+54\,e\,f^{3/2}\,x^3-16\,d^2\,\sqrt {f}\,x^2}{8\,d^{3/2}\,e+16\,d^{3/2}\,f\,x^3-54\,\sqrt {d}\,e\,f\,x^2}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 66, normalized size = 1.65 \[ - \frac {\sqrt {\frac {1}{d f}} \log {\left (- d x^{2} \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} + \frac {\sqrt {\frac {1}{d f}} \log {\left (d x^{2} \sqrt {\frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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